Introduction to Character Theory and the McKay Conjecture

1 July 11, 2016 We use the notation x = g−1xg. Theorem 1.1. Let G be a finite group, and p be a prime. Suppose further that P is a p-group that acts by automorphisms on G, and p |G|. Consider the P -invariant classes of G: clP (G) = {K : K is a conjugacy class of G and K = K for all x ∈ P} Let C = CG(P ) be the fixed point subgroup of G under the action of P . Then the map clP (G) −→ cl(C) : K 7→ K ∩ C is a well-defined bijection. First, a small lemma. Lemma 1.2. Suppose G = N oH. Then NN (H) = CN (H). Proof. Clearly CN (H) ⊆ NN (H). Conversely, take n ∈ NN (H), and h ∈ H. Then since n−1h−1n ∈ H since n normalizes H, and since h−1nh ∈ N since N EG, we see [n, h] = n−1h−1nh ∈ N ∩H = 1